Arzela-Ascoli compactness theorem.

efinition

(Uniformly equicontinuous
sequence) The sequence of functions
is "uniformly equicontinuous" if

Proposition

(Arzela-Ascoli compactness
criterion). Suppose the sequence of functions
is uniformly
bounded:
and uniformly equicontinuous then there exists a subsequence
and a continuous function
such that
converges to
uniformly on every compact subset of
.